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Estimation of Size of Filter Inductor and in 6-Pulse and 12-Pulse Bridge Praneydeep Rastogi#, Mangesh Borage*, Vineet Kumar Dwivedi*, Alok Singh*, Sunil Tiwari* and I. Annapoorani# #SELECT, VIT University, Chennai, 600 127, India *Raja Ramanna Centre for Advanced , Indore 452013, India

Abstract— For high-power mode converters operating on power supply cabinet. In this situation it would be useful if the three-phase ac mains, a rectifier with LC low pass filter is an designer could estimate the size of the inductor and capacitor. important component that provides intermediate dc link . Besides, within the given constraints, it is also possible for the Since filter inductor and capacitor constitute bulkier components designer to vary the values of the inductor and capacitor to of the overall converter, it is pertinent to estimate their sizes and optimize the filter for size, cost, weight etc. possibly minimize the overall size. The paper provides closed form expressions to estimate the size of the filter as a function of output A guideline to estimate the size of the filter inductor has power based on the previously available guidelines and datasheet been described in [5] in detail. In the first part of this paper, an values. It is also seen that it is possible to minimize overall size of expression to estimate the size of the inductor is derived using filter. While the analytical results are presented for the 6-pulse the guidelines and various values given in [5]. In the second part and 12-pulse rectifier, the validity has been confirmed by comparing the dimensions of inductor for a 1 kW, of the paper, attempt has been made to model the size of the 6-pulse rectifier. capacitor in the form of curve-fit equation using the datasheet values of commercially available [6]. A way to Keywords— Rectifier, Filter, Size, Optimization minimize the overall size of the filter is subsequently discussed. While the analytical results are presented for the 6-pulse and 12- I. INTRODUCTION pulse rectifier, the validity has been confirmed by comparing the High power switch-mode power supplies (SMPS) operate on dimensions of prototype filter inductor for a 1 kW 6-pulse three-phase ac mains. A three-phase rectifier with LC filter is rectifier. therefore invariably the front-end stage in any off-line SMPS architecture. Most commonly, a 6-pulse diode bridge rectifier [1]-[3] is used as the front-end rectifier. For high power II. DESIGN OF LC FILTER FOR 6 PULSE AND 12 PULSE applications, higher-pulse , such as 12-pulse rectifier RECTIFIER [1]-[3] are used to reduce harmonics on the source and load side.

To reduce residual harmonics on the dc side, the rectifiers are A. LC Filter for 6 Pulse Rectifier invariably associated with LC filter.[8]-[12] A 6-pulse rectifier with LC filter is shown in Fig. 1. The value of filter is chosen to maintain continuous conduction The choice of filter inductance is done to maintain mode till some value of the load current, called the critical continuous inductor current with as low as reasonably current, I . It can be obtained as: achievable to improve the load regulation of the output voltage, cri reduce harmonics in the input current and to improve the [1]. The filter capacitor is chosen to reduce the ripple in � the output voltage. Often, passive damping is incorporated in the . � = filter stage to damp the harmful in the filter circuit [4]. �� However, the study presented in this paper is limited to un- damped LC filter. The same, however, can be extended to damped filter configurations. The inductor and capacitor in the filter circuit are the bulkier components and therefore play a major part in deciding the overall size of the converter. A designer is normally provided with the specifications of a power converter and is required to provide an estimate floor space area or size of the

#Praneydeep Rastogi carried out M. Tech. project at Power Converter Development Section, Power Supplies and Industrial Accelerator Division of Raja Ramanna Centre for Advanced Technology, Indore during November Fig. 1: of 6-pulse diode rectifier. 2014 to May 2015. where,VLLmax is maximum line to line voltage, f is line where, V0 is the ripple in the output voltage, is the in Hz. ripple in the unfiltered rectified voltage and fr-12p is the ripple Δ Δ�_ The estimation of value of filter is dependent on frequency. For 12-pulse rectifier fr-12p=12f. Having chosen fc-12p, the value of filter capacitance can be determined as: the cut off frequency fc-6p which is dependent upon the ripple attenuation as:

� = _ � _ � Δ �_ = �_ III. VOLUME OF INDUCTOR where,Δ V0 is the ripple in the output voltage, is the ripple in the unfiltered rectified voltage and fr-6p is the ripple A. Size of Filter Inductance for 6 Pulse Rectifier �_ frequency. Δ For 6 pulse rectifier fr-6p=6f. Having chosenΔ fc, the It has been established in [5] that the volume of an inductor value of filter capacitance can be determined as: is a function of many parameters such as inductance value, current, operating density, current density, temperature rise, core configuration etc. In this section, the derivation of the size

of an inductor as a function of rectifier design parameters such � = as output power of the rectifier and the ratio of the critical � _� current for continuous operation to the maximum load current is

B. LC Filter for 12 Pulse Rectifier obtained. A 12-pulse series-connected diode rectifier with LC filter is shown in Fig. 2An expression to find the value of filter inductance in a 12-pulse rectifier can be derived as: . . = �

where, Kvol is the constant related to the core configuration .� � = (values available in [5]) and Ap is the area product which is the �� product of the window area and core area. The estimation of value of filter capacitance is dependent on the cut off frequency fc-12p which is dependent upon the ripple Further, the area product can be expressed as a function of attenuation as: stored in the inductor as [5],

_ +� Δ �� = = � Δ�_ �_ where, Bm is maximum density, Kj is constant related to chosen current density and temperature rise, Ku is utilization factor and x is constant related to core configuration. Values of these constants are available in [5].If Idc is the maximum dc current flowing through the inductor, then the 2 energy stored is 0.5*L*Idc .The expression for volume of filter inductor in 6-pulse rectifier can therefore be derived from (1), (7) and (8) as:

. +� � .� = [ � ] � � ⁄ where, Pd in Watts is the output power.Similarly, the expression for volume of filter inductor in 12-pulse rectifier can therefore be derived from (4), (7) and (8) as:

Fig. 2: Circuit diagram of 12-pulse series-connected diode . rectifier. +� . ∗ � � = [ ] � ��⁄ Therefore, size of inductor is dependent upon the rated power Table 1: Curve-fit constants a and b which provides the direct estimation of volumetric size of the inductor. The factors which decide the size of the filter in case Life of capacitor (hours) a b of 12 pulse rectifier is same as in case of 6 pulse diode bridge rectifier. >6000 3.2433 0.8269

IV. VOLUME OF CAPACITOR >3000,8000,10000,20000 30.912 0.5782 Electrolytic capacitors are commonly used as the filter capacitor in the front-end mains rectifier circuits. While [5] , wherein the symbols carry the usual meaning, and using provides a guideline to estimate the size of filter inductor, the (1),(3), (11) and the values of a and b for capacitors with life > authors are unaware of the procedure or an empirical 6000 hours, the expression for the volume of the filter capacitor relationship that models the size of filter capacitor. Described in for a 6-pulserectifier as a function of the output dc power, can this section, therefore, is an attempt to model the size of filter be derived as: capacitor is presented based on the size of commercially available sizes of the capacitor. Two approaches have been proposed in [7] to do that – the one that used a relationship between the size of capacitor as a function of its value and the . . other one that models the size of the capacitor as a function of � . energy stored. In this paper, the latter approach is described. � = . [ ( ) �] � _ The size of commercially available electrolytic capacitors Similarly, the expression for the volume of the filter capacitor depends on many factors: value, voltage rating, maximum for a 12-pulse rectifier as a function of the output dc power can datasheet temperature, type of construction, ripple current be derived using (4), (6) and (11) as: rating, manufacturer etc. Therefore, perhaps it is difficult to obtain a unique relationship that would describe the volume of the capacitor as a function of energy stored. As an illustrative . . example, the datasheet values of [6] are used to model the � . � . volume – energy dependence: = [ _ ( ) � ] � Therefore, the size of capacitance can be easily seen as a quantity dependent upon the rated power output. �[cm ] = ���[] where, a and b are curve-fit constants. For various capacitors of V. SIZE OPTIMIZATION OF LC FILTER [6], the volume energy dependence is shown in Fig. 3 and the – corresponding curve-fit values obtained are listed in Table 1. It A. 6-Pulse rectifier is observed, in this particular case, that two different curve-fit The total size of filter for a 6-pulse rectifier can be found possibilities are obtained for different classes of the using (9) and (12) as: capacitors.The voltage rating of the capacitor is normally chosen higher than the maximum dc-link voltage as a margin of safety, say by a factor K . Knowing that the energy stored in a capacitor v � . is +� .� = [ ] ��⁄ .. . . � . + [ _ ( ) � ] � which can be rewritten as for x = -0.125 [5]:

_� . wherein,� = . + _�� �

. . . _� � (16) = [�����] � . . .� � . (17) _� . = � (��) �

Fig. 3: Variation of capacitor volume as a function of energy for the capacitors of [6]. (18) �� _� . A proper choice of α is thus important since the volume of wherein, � � = � = . + _�� inductor increases and volume of capacitor decreases with � decrease in α, and vice versa. It is, therefore, expected that there . exists a particular value of α for which the overall size of the . _� � . filter is minimum. This value is termed as the optimum value of = [ ] � � . . α, the αopt_6p. It can be derived from (15) as: .KV f . _� . = p P (24) �� f f . � _� � = The optimum value of α, the αopt_12p. It can be derived from (21) _� � = _� as: The variation of the inductor, capacitor and total volume with respect to ratio of critical current to the rated current is plotted illustratively as shown in Fig. 3 with the following parameters . for EI lamination core type: _� _� � = _� 9 The variation of the inductor, capacitor and total volume with respect to ratio of critical current to the rated current is plotted = − . , = . , illustratively as shown in Fig. 3 with the following parameters = . , = (corresponding to ∆� = ), for EI lamination core type: � � = . �, = , = . , = , 9 � = , = . = − . , = . , = . , = (corresponding to ∆� = ), � = . �, = , = . , � = , � = , = .

Fig. 3: Variation of filter inductor, filter capacitor and total volume as a function of α for a 6-pulse rectifier, showing the existence of αopt_6p.

Fig. 4: Variation of filter inductor, filter capacitor and total

B. 12-Pulse rectifier volume as a function of α for a 12-pulse rectifier, showing the The total size of filter for a 12-pulse rectifier can be found existence of αopt_12p. using (10) and (13) as:

VI. PROTOTYPE VALIDATION OF FILTER FOR 6-PULSE RECTIFIER � . For a 6-pulse rectifier of output power 1000 W with 40 V, +� 25 A output, for f=50 Hz, VLLmax= 38 V (considering design .� = [ ] margins for various drops and line variation), fc-6p=30 Hz, ��⁄ .. . Icri=2.5 A, the values of inductance can be found out as � . L6p=630µH. . o + [ _� ( ) � ] � An inductor is fabricated for temperature rise of 25 C in EI lamination core. The design and fabrication data of the inductor which can be rewritten for x = -0.125 [5] as: is summarized in table 2. Photograph of the inductor is shown validity is confirmed with a prototype fabricated filter inductor in Fig. 5. for a 1 kW rectifier and results are encouraging. The model derived datasheet values of the electrolytic capacitors is Table 2: Summary of design and fabrication data for the approximate due to discrete values and voltage ratings available inductor from the manufacturer. It needs to be refined further with more data from different manufacturers. Parameter Value

Inductance 630 µH DC current 25 A REFERENCES Core cross sectional area 7.44 cm2 Number of turns 19 [1] Ned Mohan, T.M. Undeland,W.P.Robbins, “Power Flux density (B ) 1.2 T :Converters, Applications and Design,” November,2002,3rd m Edition, Wiley. Window utilization factor (Ku) 0.26 Dimensions of inductor 9 X 4.7 X 7.5 cm [2] M.H. Rashid, "Power Electronics: Circuits, Devices and Application” Volume of the inductor 317.25 cm3 2004,3rd Edition, Marcel Dekker.

[3] R.W. Erickson, D. Maksimovic, “Fundamentals of Power

Electronics,”2001,2nd Edition, Springer.

[4] R.D.Middlebrook, SlobodanCuk,"Advances in Switched-Mode Power Conversion,"1983, Volume I and II,2ndEdition, TESLAco.

[5] Colonel W.T. McLyman, " and Inductor Handbook,” February 2003,17th Reprint ,2nd Edition, PHI.

[6] Datasheet, Epcos AG,“ Aluminum Capacitors with Terminals,” http://en.tdk.eu/tdk-en/529316/products/product-catalog/aluminum- electrolytic-capacitors/capacitors-with-screw-terminals

[7] Praneydeep Rastogi, “Studies on Switch Mode Power Supplies for in Particle accelerators: Master Thesis, VIT University, Chennai, May 2015

[8] O.A. Ahmed, J.A.M . Bleijis,“Three phase passive bridge rectifier with low distortion input current and boosted dc output voltage,” Universities Power Engineering Conference, 2008, pp. 454-461, Apr. 2005.

[9] H.Ertl, J.W.Kolar, “A constant output current three phase diode bridge rectifier employing a novel electronic smoothing inductor,”IEEE Trans. onIndustrial Electron., vol. 52, no. 2, pp. 454-461, Apr. 2005. Fig. 5: Photograph of prototype filter inductor fabricated. [10] P.Bozovic, P.Pejovic, “Current injection based low harmonic three phase The following parameters can be noted from [2]: diode bridge rectifier operating in discontinuous conduction mode,” IEEE . The volume of inductor is Proceedings on Electrical Power Applications.,2005, vol. 152 no. 2, pp. therefore calculated using (9) as 329.61 cm3. The resulting error = 199-208, Mar. 2005. −., = . and = between the estimated and actual inductor size is 3.75 %. [11] L.C.G.de Freitas, M.G.Simoes, C.A.Canesin,L.C.de Freitas, “Performance Further, for these parameters the αopt_6p is found out to be Evaluation of a novel hybrid multi-pulse rectifier for utility interface of 0.1179. power electronic converters,” IEEE Trans. On Industrial Electron., vol. 54, no.6. , pp. 3030-3041, Dec. 2007. VII. CONCLUSION [12] A.Mansoor, W.M. Grady, R.S. Thallam, M.T. Doyle, S.D. Krein , M.J. The paper presents closed form expression derived to Samotyj, “ Effect of supply voltage harmonics on the input current of a estimate the size of filter inductor and capacitor in three phase single phase diode bridge rectifiers loads,”IEEE Trans. On Power rectifier circuits as a function of output power, earlier published Delivery, vol. 10, no.3, pp. 1416-1422, Jul. 1995. empirical constants and data sheet values. A method to optimize overall size of the LC filter is also proposed. The